Growth of preferential attachment random graphs via continuous-time branching processes

0.22 MB·English

by  K. B. Athreya

7 Growth of preferential attachment random graphs via
0
0 continuous-time branching processes
2

n
a
J Krishna B. Athreya1, Arka P. Ghosh2, and Sunder Sethuraman3

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2
February 2, 2008

]
R
P
. Abstract
h
at A version of “preferential attachment” random graphs, corresponding to linear
m “weights” with random “edge additions,” which generalizes some previously consid-
[ ered models, is studied. This graph modelis embeddedin a continuous-time branching

scheme and, usingthe branching process apparatus, several results on the graph model
1
v asymptotics are obtained, some extending previous results, such as growth rates for
9
a typical degree and the maximal degree, behavior of the vertex where the maximal
4
6 degree is attained, and a law of large numbers for the empirical distribution of degrees
1 which shows certain “scale-free” or “power-law” behaviors.
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7 Research supported in part by nsa-h982300510041, NSF-DMS-0504193 and NSF-
0 DMS-0608669.
/
h Keywordsandphrases: branchingprocesses, preferentialattachment, embedding,ran-
t dom graph, scale-free.
a
m Abbreviated title: Preferential attachment random graphs via branching processes
: AMS (2000) subject classifications: Primary 05C80; secondary 60J85 .
v
i 1 Departments of Mathematics and Statistics, Iowa State University, Ames, IA 50011
X
2 Department of Statistics, Iowa State University, Ames, IA 50011
r
a 3 Department of Mathematics, Iowa State University, Ames, IA 50011
Email: K.B. Athreya ([email protected]); A.P. Ghosh ([email protected]); S. Sethu-
raman ([email protected])
1 Introduction and results
Preferential attachment processes have a long history dating back at least to Yule [20]
and Simon [19] (cf. [12] for an interesting survey). Recently, Barabasi and Albert
[7] proposed a random graph version of these processes as a model for several real-
world networks, such as the internet and various communication structures, on which
there has been much renewed study (see [1], [9], [11], [15] and references therein). To
1
summarize, the basic idea is that, starting from a small number of nodes, or vertices,
one builds an evolving graph by “preferential attachment,” that is by attaching new
vertices to existing nodes with probabilities proportional to their “weight.” When the
weights are increasing functions of the “connectivity,” already well connected vertices
tend to become even more connected as time progresses, and so, these graphs can be
viewed as types of “reinforcement” schemes (cf. [17]). A key point, which makes these
graph models “practical,” is that, when the weights are linear, the long term degree
proportions are often in the form of a “power-law” distribution whose exponent, by
varying parameters, can be matched to empirical network data.
Thepurposeofthisnoteistounderstandageneralformofthelinear weights model
with certain random “edge additions” (described below in subsection 1.1) in terms of
an embedding in continuous-time branching processes which allows for extensions of
law of large numbers and maximal degree growth asymptotics, first approached by
difference equations and martingale methods, in [8], [10], [13], [14].
We remark some connections to branching and continuous-time Markov processes
have also been studied in two recent papers. In [18], certain laws of large numbers
for the degree distributions of the whole tree, and as seen from a randomly selected
vertex are proved for a class of “non-explosive” weights including linear weights. In
[16], asymptotic degree distributions undersuper-linear weights are considered. In this
context, the embedding given here is of a different character with respect to Markov
branching systems with immigration, and the contributions made are also different,
concentrating on detailed investigations of a generalized linear weights degree land-
scape.
1.1 Model
Start with two vertices v , v and one edge joining them–denote this graph as G . To
1 2 0
obtain G , create a new vertex v , and join it a random number X times to one of v
1 3 1 1
and v of G with equal probability. For any finite graph G = {v ,v ,...,v }, let
2 0 n 1 2 n+2
thedegreeofeachvertexbedefinedasthenumberofedgesemanatingfromthatvertex,
and the degree of the jth vertex, v ∈ G be denoted by d (n) for j = 1,...,n+ 2
j n j
and n ≥ 0 (note that in our notation, G has n+ 2 vertices at step n ≥ 0). After
n
n+2 vertices are created, to obtain G from G , create an (n+3)rd vertex v ,
n+1 n n+3
and connect it a random number X times to one of the n + 2 existing vertices
n+1
v ,...,v with probability
1 n+2
d (n)+β
i
(1.1)
n+2(d (n)+β)
j=1 j
of being joined to vertex vi for 1 ≤Pi ≤ n+2 where β ≥ 0 is a parameter. We will
also assume throughout that {X } are independently and identically distributed
i i≥1
positive integer valued random variables with distribution {p } with finite mean.
j j≥1
The “weight” then of the ith vertex at the nth step is proportional to d (n)+β, and
i
linear in the degree.
2
We remark this basic model creates a growing “tree” with undirected edges, and
includes the “one-edge” case of the original Barabasi-Albert process, made precise in
[8], by setting X ≡ 1 and β = 0, as well as the “β ≥ 0” scheme considered in [13] and
i
[14], by taking X ≡ 1. Also, the “β ≥ 1” linear case considered in [18] is recovered by
i
taking X ≡ 1.
i
The aspect of adding a random number of edges {X } at each step to vertices
i i≥1
chosen preferentially seems to be a new twist on the standard model which can be
interpreted in various ways. The results, as will be seen, involve the mean number
jp of added edges, indicating a sort of “averaging” effect in the asymptotics.
j
We also note, in the case β = 0, a more general graph process, allowing cycles and
P
self-loops, can be formed in terms of the “tree” model above (cf. [8] and Ch. 4 [11])
where several sets of edges are added to possibly different existing vertices at each step
preferentially. Namely, let {L } be independent and identically distributed positive
i i≥1
integer valued random variables with distribution {q } with finite mean, and let
j j≥1
L¯ = i L for i ≥ 1. As before, initially, we start with two vertices, v(L) and v(L)
i k=1 k 1 2
and one edge between them. Run the “tree” model now to obtain vertices {w } and
i i≥3
P
identify sets
{w ,...,w }, {w ,...,w }, ..., {w ,...,w }, ...
3 2+L1 3+L1 2+L¯2 3+L¯k−1 2+L¯k
(L) (L) (L) (L)
as vertices v ,v ,...,v ,.... One interprets the sequence of graphs G =
3 4 k+2 n
(L) (L)
{v ,...,v } for n ≥ 0 as a more general graph process where L sets of edges
1 n+2 i
are added at the ith step preferentially for i ≥ 1. This model has some overlap with
the very general model given in [10] where vertices can be selected preferentially or at
random; in[10], whenonly “new”vertices areselected preferentially, their assumptions
become X ≡ 1 and {q } has bounded support (as well as β = 0).
i j j≥1
For the remainder of the article, we will focus, for simplicity, on the basic “tree”
modelgiven through (1.1), although extensions to the othercase (L ≥ 1, β = 0) under
i
various conditions on {q } are possible.
j j≥1
1.2 Results
For n ≥ 0 and j ≥ 1, let
n+2
R (n) = I(d (n)= j)
j i
i=1
X
be the number of vertices in G with degree j. Also, define the maximum degree in
n
G by
n
M = max d (n).
n i
1≤i≤n+2
In addition, denote the mean
m = jp .
j
j≥1
X
3
Our first result is on the growth rates of individual degree sequences {d (n)}
i n≥0
and the maximal one M . It also describes the asymptotic behavior of the index where
n
the maximal degree is attained.
Theorem 1.1 Suppose (jlogj)p <∞, and let θ = m/(2m+β).
j
(i) For each i ≥ 1, there exists a random variable γ on (0,∞) such that
i
P
d (n)
i
lim = γ exists a.s..
n→∞ nθ i
(ii) Further, there exist positive absolutely continuous independent random variables
{ξ } with E[ξ ] < ∞, and a random variable V on (0,∞) such that γ = ξ V for
i i≥1 i i i
i≥ 1. In particular, for all i,j ≥ 1,
d (n) ξ
i i
lim = exists a.s..
n→∞dj(n) ξj
(iii) Also, when jrp < ∞ for an r > θ−1 = 2+β/m, then
j
P Mn
lim = maxγ < ∞ a.s.
n→∞ nθ i≥1 i
(iv) Moreover, in this case ( jrp < ∞ for r > θ−1), if I is the index where
j n
P d (n) = M ,
In n
then lim I = I < ∞ exists a.s.
n→∞ n
Remark 1.1 Note that Theorem 1.1 asserts that the individual degrees d (n) and the
i
maximal degreeM grow atthesame ratenθ, andalso thevertex with maximaldegree
n
freezes eventually, that is it does not change for large n.
The next result is on the convergence of the empirical distribution of the degrees
{d (n) : 1 ≤ i ≤ n + 2}. Let {D(y) : y ≥ 0} be a Markov branching process with
i
exponential(1)lifetimedistribution,offspringdistribution{p′ = p } ,immigration
j j−1 j≥2
rate β ≥ 0, immigration size distribution {p } , and initial value D(0) distributed
j j≥1
according to {p } (see Definition 2.2 in section 2 for the full statement). Also, for
j j≥1
y ≥ 0 and j ≥ 1, let
p (y) = P D(y) = j . (1.2)
j
(cid:16) (cid:17)
Theorem 1.2 Suppose (jlogj)p <∞, and define the probability {π } by
j j j≥1
∞
P
π = (2m+β) p (y)e−(2m+β)ydy.
j j
Z0
Then, for j ≥ 1, we have
R (n)
j
→ π , in probability, as n → ∞.
j
n
4
Remark 1.2 As a direct consequence, for bounded functions f :N → R,
∞ ∞
1
f(j)R (n) → f(j)π , in probability, as n → ∞.
j j
n
j=1 j=1
X X
We now consider the“power-law” behavior of thelimit degree distribution{π } .
j j≥1
Theorem 1.3 Suppose j2+β/mp < ∞. Then, for s ≥ 0, we have
j≥1 j
jsπP< ∞ if and only if s< 2+β/m.
j
j≥1
X
Remark 1.3 Heuristically, the last result suggests π = O(j−[3+β/m]) as j ↑ ∞. In
j
the case X ≡ x for x ≥ 1, (1.2) can be explicitly evaluated (Proposition 3.2) to get
i 0 0
π = O(j−[3+β/x0]) when j is a multiple of x .
j 0
The next section discusses the embedding method and auxilliary estimates. In the
third section, the proofs of Theorems 1.1, 1.2, and 1.3 are given.
2 Embedding and some estimates
We start with the following definitions, and then describe in following subsections the
embedding and various estimates.
Definition 2.1 A Markov branching process with offspring distribution {p′} and
j j≥0
lifetime parameter 0 < λ < ∞ is a continuous-time Markov chain {Z(t) : t ≥ 0} with
state space S = {0,1,2...} and waiting time parameters λ ≡ iλ for i ≥ 0, and jump
i
probabilities p(i,j) = p′ for j ≥ i−1 ≥ 0 and i ≥ 1, p(0,0) = 1, and p(i,j) = 0
j−i+1
otherwise (cf. Chapter III [5]).
Definition 2.2 A Markov branching process with offspring distribution {p′} and
j j≥0
lifetime parameter 0 < λ < ∞, immigration parameter 0 ≤ β < ∞ and immigration
size distribution {p } is a continuous-time Markov chain {D(t) : t ≥ 0} such that
j j≥0
D(t)= Z(t) as in Definition 2.1 when β = 0, and when β > 0,
∞
D(t) = Z (t−T ) I(T ≤ t)
i i i
i=0
X
where {T } are the jump times of a Poisson process {N(t) : t ≥ 0} with parameter
i i≥1
β, T = 0, and {Z (·)} are independent copies of {Z(t) : t ≥ 0} as in Definition
0 i i≥0
2.1, with Z (0) = D(0) and Z (0) distributed according to {p } for i ≥ 1 and also
0 i j j≥0
independent of {N(t) :t ≥ 0}.
Remark 2.1 The condition that the mean number of offspring is finite, jp′ < ∞,
j
is sufficient to ensure that P(Z(t) < ∞) = 1 and P(D(t) < ∞) = 1 for all t ≥ 0, that
P
is no explosion occurs in finite time (cf. p. 105 [5])
5
2.1 Embedding process
We now construct a Markov branching process through which a certain “embedding”
is accomplished. Recall {p } is a probability on the positive integers. Consider an
j j≥1
infinitesequenceof independentprocesses{D (t) : t ≥ 0} whereeach {D (t) :t ≥ 0}
i i≥1 i
is a Markov branching process with immigration as in Definition 2.2, corresponding to
exponential(λ = 1) lifetimes, offspring distribution {p′ = p } (with p′ = p′ = 0),
j j−1 j≥2 0 1
and immigration parameter β ≥ 0 and immigration size distribution {p } . The
j j≥1
distributions of {D (0)} will be specified later.
i i≥1
Now, define recursively the following processes.
• At time 0, the first two processes {D (t) : t ≥ 0} are started with D (0) =
i i=1,2 1
D (0) = 1. Let τ = τ = 0, and τ be the first time an “event” occurs in any one of
2 −1 0 1
the two processes.
•Now addarandomX′ ofnewparticles totheprocessinwhichtheeventoccurred:
1
(i) If the event is “immigration,” then P(X′ = j) = p for j ≥ 1. (ii) If the event is the
1 j
death of a particle, then P(X′ = j) = p for j ≥ 2. Denote X as the net addition;
1 j−1 1
then P(X = j) = p for j ≥ 1.
1 j
• At time τ , start a new Markov branching process {D (t) : t ≥ 0} with D (0) =
1 3 3
X .
1
•Letτ bethefirsttimeafterτ thataneventoccursinanyoftheprocesses{D (t) :
2 1 i
t ≥ τ } and {D (t−τ ) : t ≥ τ }. Add a random (net) number X , following the
1 i=1,2 3 1 1 2
scheme above for X , of particles with distribution {p } to the process in which the
1 j j≥1
event occurred. At time τ , start a new Markov branching process{D (t) : t ≥ 0} with
2 4
D (0) = X .
4 2
• Supposethat n processes have been started with the first two at τ = 0, the third
0
at time τ , the fourth at time τ , and so on with the nth at time τ , and with (net)
1 2 n−2
additions X ,X ,...,X at these times. Now, let τ be the first time after τ
1 2 n−2 n−1 n−2
that an event occurs in one of the processes {D (t) :t ≥ 0} , {D (t−τ ) : t ≥ τ },
i i=1,2 3 1 1
{D (t−τ ) :t ≥ τ },...,{D (t−τ ) :t ≥ τ }. Add a (net) random number X
4 2 2 n n−2 n−2 n−1
of new particles with distribution {p } (following the scheme above) to the process
j j≥1
in which the event happened. Now start the (n+1)st process {D (t) : t ≥ 0} with
n+1
D (0) = X .
n+1 n−1
Theorem 2.1 [Embedding Theorem] Recall the degree sequence d (n) defined for
j
the graphs {G } near (1.1). For n≥ 0, let
n
Z ≡ {D (τ −τ ) :1 ≤ j ≤ n+2}, and
n j n j−2
Z˜ ≡ {d (n): 1≤ j ≤ n+2}.
n j
Then, the two collections {Z } and {Z˜ } have the same distribution.
n n≥0 n n≥0
Proof. First note that both sequences {Z } and {Z˜ } have the Markov
n n≥0 n n≥0
property and Z = Z˜ = {1,1}. Next, it will be shown below that the transition
0 0
6
probability mechanism from Z to Z is the same as that from Z˜ to Z˜ . To see
n n+1 n n+1
this note that, at time 0, both D (·) and D (·) are “turned on,” and, at time τ , D (·)
1 2 1 3
is “turned on,” and more generally, at τ , D (·) is “turned on.” At time τ , the
j j+2 n+1
“event” could be in D (·) for 1≤ i ≤ n+2 with probability
i
D (τ −τ )+β
i n i−2
n+2(D (τ −τ )+β)
j=1 j n j−2
inviewofthefactthattheminPimumofn+2independentexponentialrandomvariables
{η } with means {µ−1} is an exponential random variable with mean
i 1≤i≤n+2 i 1≤i≤n+2
( n+2µ )−1, and coincides with η with probability µ ( n+2µ )−1 for 1 ≤ i ≤ n+2.
i=1 i i i i=1 i
At that event time τ , D (·) is “turned on,” that is a new (n + 3)rd vertex is
n+1 n+3
P P
created and connected to the chosen vertex v with X edges between them. Hence
i n+1
both the degree of the new vertex and increment in the degree of the chosen vertex
(among the existing ones) is X . This shows that the conditional distribution of
n+1
Z given Z = z is the same as that of Z˜ given Z˜ = z. (cid:3)
n+1 n n+1 n
2.2 Estimates on branching times
Wenowdevelopsomepropertiesofthebranchingtimes{τ } ,usedintheembedding
n n≥1
insubsection2.1, whichhavesomeanalogytoresultsinsectionIII.9[5](cf. [4]). Define
S = 2+2β and, for n ≥ 1,
0
n
S = 2+2β + 2X +nβ,
n j
j=1
X
where as before X ,...,X are the independent and identically distributed according
1 n
to {p } net additions at event times τ ,...,τ .
j j≥1 1 n
Proposition 2.1 The random variable τ is exponential with mean S−1. Also, for
1 0
n ≥ 1, conditioned on the σ-algebra F generated by {D (t − τ ) : τ ≤ t ≤
n j j−2 j−2
τ ;X } , the random variable τ −τ is exponential with mean S−1.
n j 1≤j≤n n+1 n n
Proof. Follows from the construction of the {τ } . (cid:3)
i i≥1
Proposition 2.2 Suppose m = jp < ∞. Then,
j
P n
1
τ − ;F
n n
S
(cid:26) j=1 j−1 (cid:27)n≥1
X
is an L2 bounded martingale and hence converges a.s. as well as in L2.
7
Proof. The martingale property follows from the fact
n
τ = (τ −τ )
n j j−1
j=1
X
and Proposition 2.1.
Next, with φ(a) = E[e−aX1] for a≥ 0, we have the uniform bound in n≥ 1,
n n
1 1
Var τ − = Var τ −τ −
n j j−1
S S
(cid:18) j=1 j−1(cid:19) (cid:18)j=1(cid:18) j−1(cid:19)(cid:19)
X X
n
1
= Var τ −τ − (by martingale property)
j j−1
S
j=1 (cid:18) j−1(cid:19)
X
n
1
= E
S2
j=1 (cid:20) j−1(cid:21)
X
n ∞
= E xe−Sj−1xdx
j=1 (cid:20)Z0 (cid:21)
X
∞ ∞ j−1
≤ φ(2x)e−xβ xe−(2+2β)xdx
j=1Z0 (cid:18) (cid:19)
X
∞ xe−(2+2β)xdx
≤ < ∞
1−φ(2x)e−xβ
Z0
where the finiteness in the last bound follows from the fact that
x 1
lim = < ∞.
x↓0 1−φ(2x)e−xβ 2m+β
Thea.s. and L2-convergence follows from Doob’s martingale convergence theorem (c.f.
Theorem 13.3.9 [6]). (cid:3)
Proposition 2.3 Suppose (jlogj)p < ∞, and recall m = jp . Let also α =
j j
(2m+β)−1. Then, there exists a real random variable Y so that a.s.,
P P
n
α
lim τ − = Y.
n
n→∞ j
j=1
X
Proof. By Proposition 2.2, there is a real random variable Y′ such that,
n
1
τ − → Y′ a.s.
n
S
j−1
j=1
X
To complete the proof, we note, as E[X logX ] = (jlogj)p < ∞, by Theorem
1 1 j
III.9.4 [5] on reciprocal sums, that ∞ (1/S −α/j) converges a.s. (cid:3)
j=1 j P
P
8
Corollary 2.1 Suppose m = jp < ∞. Then,
j
(i) τ ↑∞ a.s., as n→ ∞.
n
P
Also, when (jlogj)p < ∞, we have, with α= (2m+β)−1, that
j
(ii) τ −αlogn→ Y˜ := Y −αγ a.s., as n → ∞, where γ is the Euler’s constant.
n
P
(iii) For each fixed ǫ > 0, sup (τ −τ −αlog(n/k)) → 0 a.s., as n → ∞.
nǫ≤k≤n n k
Proof. The first claim follows from Proposition 2.2 and the fact that 1/S = ∞,
j
sincebystronglaw of largenumbers,wehave a.s. thatS ≤ j(1/α+1) forlarge j. The
j
last two claims, as n 1/j −logn → γ, Euler’s constant, are direct conPsequences of
j=1
Proposition 2.3. (cid:3)
P
2.3 Estimates on Markov branching processes
As in Definition 2.2, let {D(t) : t ≥ 0} be a Markov branching process with offspring
distribution {p′ = p } , lifetime λ = 1 and immigration β ≥ 0 parameters, and
j j−1 j≥2
immigration distribution {p } .
j j≥1
Proposition 2.4 Suppose (jlogj)p < ∞, and D(0) ≥ 1, E[D(0)] < ∞. Recall
j
m = jp . Then,
j
P
lim D(t)e−mt = ζ
P t→∞
converges a.s. and inL1, and ζ issupported on (0,∞) and has anabsolutely continuous
distribution.
Proof. Let β > 0; when β = 0 the argument is easier and a special case of the
following development. Let 0 = T < T < ···T < ··· be the times at which immi-
0 1 n
gration occurs, and let η ,η ,... be the respective number of immigrating individuals
1 2
(distributed according to {p } ). From Definition 2.2, D(t) has representation
j j≥1
∞
D(t) = Z (t−T ) I(T ≤ t) (2.1)
i i i
i=0
X
where {Z (t) : t ≥ 0} are independent Markov branching processes with offspring
i i≥0
distribution {p′ = p } , with exponential(λ = 1) lifetime distributions, with no
j j−1 j≥2
immigration, with Z (0) = D(0) and Z (0) = η for i ≥ 1, and also independent of
0 i i
{T } . Under the hypothesis (jlogj)p < ∞, it is known (Theorem III.7.2 [5];
i i≥0 j
with rate λ( jp′ −1) = (j +1)p −1 = m), for i≥ 0, that
j≥2 j j≥P1 j
P P lim Z (t)e−mt = W (2.2)
i i
t→∞
converges in (0,∞) a.s. and W has a continuous distribution on (0,∞). Also under
i
the hypothesis that (jlogj)p < ∞, it can be shown (Proposition 2.5) that
j
PE[W ]< ∞ where W = supZ (t)e−mt, (2.3)
i i i
t≥0
f f
9
and hence convergence in (2.2) holds in L1 as well.
Since {T } is a Poisson process with rate β, and independent of {Z (t)} ,
i i≥0 i t≥0
∞ ∞ i
β
E W e−mTi ≤ E[W ] E[D(0)]+ < ∞, (2.4)
i 1
m+β
(cid:20)i=0 (cid:21) (cid:18) i=1(cid:18) (cid:19) (cid:19)
X X
f f
yielding
∞
W e−mTi < ∞ a.s.. (2.5)
i
i=0
X
Hence, noting (2.2), (2.3) and (2.5f), by dominated convergence,
∞
lim D(t)e−mt = lim Z (t−T )I(T ≤ t)e−m(t−Ti) e−mTi
i i i
t→∞ t→∞
i=0 (cid:20) (cid:21)
X
∞
= W e−mTi := ζ (2.6)
i
i=0
X
converges in (0,∞) a.s.. Also,
∞
supD(t)e−mt ≤ W e−mTi (2.7)
i
t≥0
i=0
X
f
and hence by (2.4) and (2.6), we get that
lim D(t)e−mt = ζ in L1.
t→∞
Finally, since{W } ,{T } areindependent,absolutelycontinuous randomvari-
i i≥0 i i≥1
ables, ζ is absolutely continuous as well. (cid:3)
2.4 Suprema estimates
We give now some moment estimates which follow by combination of results in the
literature. Let{Z(t) :t ≥ 0}beaMarkov branchingprocesswithoffspringdistribution
{p′ = p } and lifetime parameter λ = 1 as in Definition 2.1 with independent
j j−1 j≥2
initial population Z(0) distributed according to {p } . Recall m = jp , and, from
j j≥1 j
(2.2) and (2.3), that
P
W = lim Z(t)e−mt and W = supZ(t)e−mt.
t→∞ t≥0
f
10